📂Lebesgue Spaces

Eigenvalues and Eigenfunctions in S-L Problems

Definition¹

If the Sturm-Liouville differential equation

$$\begin{equation} \left[ p(x)u^{\prime}(x) \right]^{\prime}+\left[ q(x) +\lambda w(x) \right]u(x)=0 \end{equation}$$

has a solution $u \in L_{r}^{2}(a,b)$ different from $0$, then $\lambda$ is called an eigenvalue, and the corresponding $u$ is referred to as the eigenfunction.

Explanation

Let’s assume the weighting function is $w(x)=1$. Then, $(1)$ can be written as follows.

$$\begin{equation} \begin{aligned} && p(x)u^{\prime \prime}(x) +p^{\prime}(x)u^{\prime}(x)+q(x)u(x)+\lambda u(x) =&\ 0 \\ \implies && -p(x)u^{\prime \prime}(x) -p^{\prime}(x)u^{\prime}(x)-q(x) =&\ \lambda u(x) \end{aligned} \end{equation}$$

Let’s suppose the operator $D:C^{2}[a,b] \to C[a,b]$ is as follows.

$$Du(x):=-p(x)\frac{ d ^{2}u(x)}{ d x^{2} }-p^{\prime}(x)\frac{ d u(x)}{ d x }-q(x)u(x)$$

Then, $(2)$ can be expressed as below.

$$Du =\lambda u$$

In the S-L problem, $\lambda$ becomes the eigenvalue, and $u$ is the corresponding eigenfunction.